I'm trying to walk a fine line. On the one hand the 12 note equal tempered scale has almost surely a societal component to it. That is, it was one that won out for weird cultural reasons that don't have a lot to do with some idea of correctness or utility (like why 'qwerty' won out over Dvorak keyboards).
But there is an explanation that does try and get at it a bit more analytically and that's in the paper I linked to. With the above caveats about the cultural momentum, the 12 not equal tempered scale provides a happy compromise between the number of notes to provide a basic building block for music (characters or digits would be an analogy to notes in an octave) vs the number of "good" note pair combinations (where "good" is if the frequencies have a small/simple fraction approximation).
Some of it is hand-waivey, to be sure, but at least it provides a potential reason and a starting point.
As far as I can tell, 12 ET is the smallest equal temperament that acceptably encodes the common variants of pentatonic scale. Meanwhile, the pentatonic scale probably appears because (3/2)^5 ≈ 8.
But there is an explanation that does try and get at it a bit more analytically and that's in the paper I linked to. With the above caveats about the cultural momentum, the 12 not equal tempered scale provides a happy compromise between the number of notes to provide a basic building block for music (characters or digits would be an analogy to notes in an octave) vs the number of "good" note pair combinations (where "good" is if the frequencies have a small/simple fraction approximation).
Some of it is hand-waivey, to be sure, but at least it provides a potential reason and a starting point.